Summary

This activity uses simulation to help students understand sampling variability and reason about whether a particular sample result is unusual, given a particular hypothesis. By using first candies, then a web applet, and varying sample size, students learn that larger samples give more stable and better estimates of a population parameter and develop an appreciation for factors affecting sampling variability.

First, students estimate the proportion of orange Reese's pieces in a random sample of 25 candies. Students then use an actual sample of Reese's pieces candy to calculate a sample proportion, and then compare results for different samples, taken by each student in the class. Next, students will use the Web Applet Reese's Pieces (at rossmanchance.com) to gather information on the sample proportions of orange candies in random samples of 25. During this stage of the activity students will compare their group and the class results to the actual parameters. In the final part of the activity, the students will use the Applet to investigate the effect of sample size on correctly estimating parameters.

Learning Goals

To understand variability between samples

To build and describe distributions of sample proportions

To understand the effect of sample size on how well a sample resembles a population

To develop an understanding of a sample distribution

To develop an understanding of a sampling distribution

To develop an understanding of a population distribution

To develop an understanding of the differences between sample, sampling, and population distributions

Context for Use

This activity is most appropriate for an introductory statistics course.

Description and Teaching Materials

This activity takes place over three stages. The following sections give the lesson details and are to be used with a student handout. (Copy of Student Handout)(Microsoft Word 93kB Feb25 07)

Stage One of Activity: Studying the variability of orange candies

Inform students that Reese's Pieces candies have three colors: orange, brown, and yellow. Ask them: Which color do you think has more candies in a package: orange, brown or yellow?

Have students guess the proportion of each color (orange, brown, and yellow) in a bag and record their answers.

Next ask students...If each student in the class takes a sample of 25 Reeses pieces, would you expect every student to have the same number of oranges in their sample?

Next have students pretend that 10 students each took samples of 25 Reese's pieces. Ask them to write down the number of oranges they might expect for these 10 samples. Tell them these numbers represent the variability you would expect to see in the number of orange candies in 10 samples of 25 pieces.

Next tell students...You will be given a cup that is a random sample of Reeses pieces. Count out 25 candies from this cup without paying attention to color. In fact, try to IGNORE the colors as you do this.

Inform students...Now, count the colors for your sample and record the number of each color as well as the proportion of each color sampled from the 25.

Have students write the number AND the proportion of orange candies in their sample on the board. Mark where each value should be on the two dotplots your teacher constructs (construct dotplots - one for number of oranges, one for proportion of oranges).

Stage Two of Activity: simulating samples using an applet

Access the Reese's Pieces Applet at Rossmanchance.com (See Available Technologies)

Tell students...Instead of trying this activity again with fewer or more candies, simulate the activity using a web applet. Go to www.rossmanchance.com/applets/, and look at the bottom of the far right column to find Java Applets. Click on Java Applets and look for Sampling Distributions, and click on Reese's Pieces. You will see a big container of colored candies: that represents the POPULATION.

Ask students: How many orange candies are in the population? You will see that the proportion of orange is already set at .45, so that is the population parameter. (People who have counted lots of Reese's pieces came up with this number).

Ask...How does .45 compare to the proportion of orange candies in your sample?

Ask students: .How does it compare to the center of the class distribution?

Click on the draw samples button. One sample of 25 candies will be taken and the proportion of means for this sample is plotted on the graph. Repeat this again.

Ask students: Do you get the same or different values for each sample?

Ask students...How do these numbers compare to the ones our class obtained?

Ask students: How close is each sample statistic (proportion) to the POPULATION PARAMETER?

Have students turn off the animation and change the number of samples to 100. Click on draw samples, and see the distribution of sample statistics built. Describe its shape, center and spread.

Ask students: How does this compare to the one our class constructed on the board?

Stage Three of Activity: studying the effect of sample size

Ask students: What happens to this distribution of sample statistics if they change the number of candies in each sample (sample size)?

Have them change the sample size to 10 and draw 100 samples. Ask them: How close is each sample statistic (proportion) to the POPULATION PARAMETER?

Next, have them change the sample size to 100 and draw 100 samples. How close is each sample statistic (proportion) to the POPULATION PARAMETER?

Teaching Notes and Tips

It is helpful for the teacher to go through the entire activity using the web applet themselves, before having students do the activity.

Be sure to let students make and discuss their conjectures before looking at real or simulated data.

Be sure to have students turn off the animation on the web applet after they have taken a few samples.

Be sure to ask students to compare their results and discuss the amount of sampling variability.

Assessment

Here are some questions that can be given to students to discuss or write answers to:

Distinguish between how samples vary from each other, and variability of data WITHIN one sample of data.

Why is that larger samples better represent the population from which they were sampled than small samples?

Based on the class activity, would you be surprised to get a sample that had only 5 orange candies? Why or why not?